Question

Determinați valorile intregi lui x pentru care fractia
$\frac{5}{2x - 1}$
este nr întreg.

Answer 1

$\frac{5}{2x - 1} = \\ \frac{5}{2 \times 3 - 1} = \frac{5}{5} = 1 \\ \frac{5}{2 \times 1 - 1} = 5$

Answer 2

$\frac{5}{2x - 1} \: nr. \: intreg \: = > 2x - 1 \: \epsilon \: D_{5}$

$D_{5} = \begin{Bmatrix} \pm1, \pm5 \end{Bmatrix}$

$2x - 1 = - 1$

$2x = - 1 + 1$

$2x = 0$

$x = \frac{0}{2}$

$x = 0$

$2x - 1 = 1$

$2x = 1 + 1$

$2x = 2$

$x = \frac{2}{2}$

$x = 1$

$2x - 1 = - 5$

$2x = - 5 + 1$

$2x = - 4$

$x = - \frac{4}{2}$

$x = - 2$

$2x - 1 = 5$

$2x = 5 + 1$

$2x = 6$

$x = \frac{6}{2}$

$x = 3$

$= > x \: \epsilon \: \begin{Bmatrix} - 2 ,0,1,3 \end{Bmatrix}$

$pt.\:x=-2=>\frac{5}{2\times{(-2)}-1}=\frac{5}{-4-1}=\frac{5}{-5}=-1\:\epsilon\:Z$

$pt.\:x=0=>\frac{5}{2\times0-1}=\frac{5}{0-1}=\frac{5}{-1}=-5\:\epsilon\:Z$

$pt.\:x=1=>\frac{5}{2\times1-1}=>\frac{5}{2-1}=\frac{5}{1}=5\:\epsilon\:Z$

$pt.\:x=3=>\frac{5}{2\times3-1}=\frac{5}{6-1}=\frac{5}{5}=1\:\epsilon\:Z$